(at around 1h 14 mins) Between when Cyclops blows open a door in the Statue of Liberty, and when he sees Jean in peril, he gains and loses a pair of gloves between shots.
In early scenes the filmmakers went to some effort to make Cyclops appear taller than Wolverine, but about half way through the film they just stopped, and suddenly Wolverine appears considerably taller than Cyclops (consistent with the difference in height of the actors themselves).
Before and during the fight at the train station, the time of day changes from dark to light several times.
(at around 1h 23 mins) During the fight atop the Statue of Liberty, Sabretooth throws Wolverine away from the statue's crown. In the next shot, their positions are reversed, with Wolverine's back to the crown.
(at around 1h 29 mins) When Jean Gray is checking Logan after the fight you can see a pink bra under her jacket. The camera then switches to Logan and when it returns to Jean Gray the bra is missing and her coat closed up.
(at around 2 mins) The Nazi Schutz-Staffel guards are wearing steel helmets. These were only used in combat situations, which did not include taking in and sorting victims for genocide. Camp guards wore "garrison caps" instead.
(at around 2 mins) The Nazi policy sorting of newly-arrived victims of genocide into two groups - one to be murdered on site and one to be enslaved - was called "selektion" and usually involved SS doctors such as Doctor Josef Mengele, who made the selection according to a particular captive's state of health as he perceived it. In the movie no such officials are present. When this scene was re-shot 11 years later for X-Men: First Class (2011) they got this right.
(at around 18 mins) When Wolverine hits the fallen tree, he and Rogue lunge forward before hitting the tree. Hitting the tree, Wolverine goes flying through the windshield and you can see the whole windshield is flying out but looking back at Rogue, it is back but cracked. In a real collision, he would have just hit the steering wheel and windshield and not fly through the windshield that far. He would need to be sitting on top of the car in order to fly out of the car that far when hitting the tree. Lastly, when Wolverine is told to wear a seat belt, they are not moving very fast but when they hit the tree, Wolverine flies out the window a distance one would need to be going over 100 to do so. If he was moving fast enough the whole front end of the truck would be crushed.
(at around 2 mins) The Nazis separated the men from the women the minute they stepped off the train once they arrived at a concentration camp. Erik and his father would have been separated from his mother long before they made it as far as they did in the film.
(at around 1 min) In the Nazi scene, the Schutz-Staffel guard in the watch tower is manning a MG-42 multipurpose machine gun, an equipment usually issued to front line troops only. For behind-the-line duty such as guarding concentration camps, obsolete weapons or those captured from occupied countries such as France or Czechoslovakia were usually issued.
Minor inconsistencies in each mutant's power (especially relating to matters never clearly defined in the comic books) are not goofs. Since it is all make believe, they can do whatever the writers say they can do, even when contradictions seem to occur.
Outside the train station, exhaust from one of the police cars can be seen moving back into the tailpipe, indicating the shot is running backwards.
(at around 1h 14 mins) In the Statue of Liberty, after Toad spits his slime on Jean Grey and he falls, you can see the rope that was holding him up and where it connects on his back.
(at around 54 mins) When Magneto has Wolverine floating in the air at the train station, and the camera zooms into Magneto bending his claws, you can see two Phillips head screws on the palm side of each claw down by his hand.
(at around 21 mins) When Magneto takes Wolverine's dog tag from around Sabretooth's neck, the clear string attached to pull it can faintly be seen.
When Rogue tells Wolverine to put on his seat belt right before Wolverine crashes his truck, actor Hugh Jackman's line is said with his normal Australian accent.
(at around 11 mins) When we see Wolverine's first full cage fight, every hit he took or made should have sounded metallic due to his adamantium enhancements, but we only hear the sound when Wolverine himself hits his opponent.
Professor Charles Xavier has always been depicted as an American, having been born in New York City, in the Marvel comics.
(at around 51 mins) When Wolverine is talking to Rogue on the train, she begins to cry and he pulls her head to his chest. In one scene you can see his lips moving (speaking the next line) without sound, until the shot changes and he speaks the line in sync.
At 1:44:25 when Wolverine is fighting Mystique, she rips off a metal gate from the metal fence and hit him with it, you can clearly see the "metal" gate bowing and bouncing a little, because it is made from rubber or something else not metal.
(at around 38 mins) The camera is visible in Cyclops' glasses when he is in Logan's room and says, "Oh, and Logan: stay away from my girl."
(at around 1h 14 mins) Just after Toad has spat on Jean Grey and fallen to the ground, the rope holding him up in the air can be seen.
(at around 46 mins) Xavier says he is unable to find Magneto's signal from Cerebro. We later see that this is because Magneto wears a stealth helmet. But Magneto's omnipresent accomplices Sabretooth, Mystique, and Toad, do not have such helmets, and no similar cloaking devices are ever mentioned. Therefore, Xavier could still see their locations and reasonably assume that Magneto is in the same general vicinity as them. It is odd that this simple logic wouldn't occur to Xavier, who is supposed to be a genius.
The sharpness of Logan's claws changes to suit the situation in different scenes. The claws easily slice though the metal wire fence when fighting Mystique but he is able to pull her towards him with the metal chain stuck on his claws without slicing through the chain. (Also, when on the Statue of Liberty his claws slice easily though the spikes on the statue's head but when he is climbing up the side with his claws they only puncture the statue they don't slice though it once they're in, even though all his body weight must be hanging on his claws alone.)
Bobby (Mystique in disguise) complains Rogue for using her power against another mutant, after she touched Wolverine for healing herself when accidentally he nailed her in the shoulder, asking her for leaving the school. It is never explained how Mystique learned about the incident between them.
(at around 1h 14 mins) During the Statue of Liberty fight, Mystique kicks Wolverine in the side of his head. Seeing as Wolverine's skeleton is laced with adamantium, it should have broken her foot. Yet, she shows no signs of injury or pain from it.
The stealth helmet that Magneto wears is supposed to protect him from Charles reading his mind or locating him with Cerebro. The helmet covers the thickest part of his skull, leaving the thinnest part fully exposed--his face. It is unreasonable to assume that this would not be a big enough area for the professor's powers to penetrate.
(at around 4 mins) Marie, in her first scene, points to a map and says "Niagara Falls" which is on the New York-Ontario border, but her finger is touching the Minnesota-Wisconsin border.
When Professor X tries to communicate with Magneto via telepathy and cannot, he mentions that 'Eric must be blocking him somehow.'. This is odd considering Magneto was first seen wearing the helmet, allegedly, in X-Men: First Class. Magneto blocked Charles's telepathy when he confronted Kevin Bacon's character on the submarine.